Long-time behavior of solutions to the generalized Allen-Cahn model with degenerate diffusivity

摘要

The generalized Allen-Calm equation, u(t) = epsilon(2)(D(u)u(x))(x) - epsilon(2)/2 D'(u)u(x)(2) - F'(u) with nonlinear diffusion, D = D(u), and potential, F = F(u), of the form D(u) = vertical bar 1 - u(2)vertical bar(m), or D(u) = vertical bar 1 - u vertical bar(m), m > 1, and F(u) = 1/2n vertical bar 1 - u(2)vertical bar(n), n >= 2, respectively, is studied. These choices correspond to a reaction function that can be derived from a double well potential, and to a generalized degenerate diffusivity coefficient depending on the density u that vanishes at one or at the two wells, u = +/- 1. It is shown that, interface layer solutions that are equal to +/- 1 except at a finite number of thin transitions of width e persist for an either exponentially or algebraically long time, depending upon the interplay between the exponents n and m. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are derived.